A131688 Decimal expansion of the constant Sum_{k>=1} log(k + 1) / (k * (k + 1)).

1, 2, 5, 7, 7, 4, 6, 8, 8, 6, 9, 4, 4, 3, 6, 9, 6, 3, 0, 0, 0, 9, 8, 9, 9, 8, 3, 0, 4, 9, 5, 8, 8, 1, 5, 2, 8, 5, 1, 1, 5, 4, 0, 8, 9, 0, 5, 0, 8, 8, 8, 4, 8, 6, 8, 9, 7, 7, 5, 4, 0, 8, 3, 3, 5, 2, 2, 5, 4, 9, 9, 9, 4, 8, 9, 3, 7, 4, 4, 9, 3, 4, 9, 7, 0, 7, 9, 0, 4, 7, 3, 1, 5, 0, 1, 9, 0, 9, 7, 8, 2, 4, 5, 4, 8

Offset

1,2

Comments

Given A131385(n) = Product_{k=1..n} floor((n+k)/k), then limit A131385(n+1)/A131385(n) = exp(c), where c = this constant. - Paul D. Hanna, Nov 26 2012

Closely related to A085361 (the exponent in the definition of A085291). - Yuriy Sibirmovsky, Sep 04 2016

References

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, page 62. [Jean-François Alcover, Mar 21 2013]

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

Khristo N. Boyadzhiev, A special constant and series with zeta values and harmonic numbers, arXiv:1903.11141 [math.NT], 2019.

Mark W. Coffey, Series of zeta values, the Stieltjes constants and a sum S_gamma(n), arXiv:math-ph/0706.0345, 2007-2009, eq (38a).

Paul Erdős, S. W. Graham, Aleksandar Ivic and Carl Pomerance, On the number of divisors of n!, Analytic Number Theory, Volume 138, Progress in Mathematics pp 337-355.

Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 538.

Sofia Kalpazidou, Khintchine's constant for Lüroth representation, Journal of Number Theory, Volume 29, Issue 2, June 1988, Pages 196-205.

Formula

Equals Sum_{s>=1} (-1)^(s+1)*zeta(s+1)/s.

Equals Sum_{k>=1} -zeta'(1 + k), where Zeta' is the derivative of the Riemann zeta function. - Vladimir Reshetnikov, Dec 28 2008

Equals Sum_{s>=1} log(1+1/s)/s. - Jean-François Alcover, Mar 26 2013

Equals Integral_{t=0..1} H(t)/t dt. Compare to A001620 = Integral_{t=0..1} H(t) dt. Where H(t) are generalized harmonic numbers. - Yuriy Sibirmovsky, Sep 04 2016

Equals lim_{n->oo} log(d(n!))*log(n)/n, where d(n) is the number of divisors of n (A000005) (Erdős et al., 1996). - Amiram Eldar, Nov 07 2020

Example

1.257746886944369630009899830495881528511540890508884868977540833522...

Maple

evalf(sum((-1)^(n+1)*Zeta(n+1)/n, n=1..infinity), 120); # Vaclav Kotesovec, Dec 11 2015
evalf(Sum(-Zeta(1, k), k = 2..infinity), 120); # Vaclav Kotesovec, Jun 18 2021

Mathematica

Sum[ -Zeta'[1 + k], {k, 1, Infinity}] (* Vladimir Reshetnikov, Dec 28 2008 *)
Integrate[EulerGamma/x + PolyGamma[0, 1+x]/x, {x, 0, 1}] // N[#, 105]& // RealDigits[#][[1]]& (* or *) Integrate[x*Log[x]/((1-x)*Log[1-x]), {x, 0, 1}] // N[#, 105]& // RealDigits[#][[1]]& (* Jean-François Alcover, Feb 04 2013 *)
$MaxExtraPrecision = 200; NIntegrate[HarmonicNumber[t]/t, {t, 0, 1}, WorkingPrecision -> 105] (* Yuriy Sibirmovsky, Sep 04 2016 *)
digits = 120; RealDigits[NSum[(-1)^(n + 1)*Zeta[n + 1]/n, {n, 1, Infinity}, NSumTerms -> 20*digits, WorkingPrecision -> 10*digits, Method -> "AlternatingSigns"], 10, digits][[1]] (* G. C. Greubel, Nov 15 2018 *)

Prog

(PARI) sumalt(s=1, (-1)^(s+1)/s*zeta(s+1) )
(PARI) suminf(k=2, -zeta'(k)) \\ Vaclav Kotesovec, Jun 17 2021
(Magma) SetDefaultRealField(RealField(100)); L:=RiemannZeta(); (&+[(-1)^(n+1)*Evaluate(L, n+1)/n: n in [1..10^3]]); // G. C. Greubel, Nov 15 2018
(SageMath) numerical_approx(sum((-1)^(k+1)*zeta(k+1)/k for k in [1..1000]), digits=100) # G. C. Greubel, Nov 15 2018

Crossrefs

Cf. A000005, A002210, A027423, A075887, A131385, A244109, A001620, A085361, A345682.

Sequence in context: A325437 A141430 A021392 * A226213 A199590 A096624

Adjacent sequences: A131685 A131686 A131687 * A131689 A131690 A131691

Keyword

nonn,cons

Author

R. J. Mathar, Sep 14 2007

Extensions

Extended to 105 digits by Jean-François Alcover, Feb 04 2013

Status

approved